The South Pacific Convergence Zone (SPCZ) is poorly represented in global coupled simulations from the Coupled Model Intercomparison Project Phase 5 (CMIP5), with trademark biases such as the tendency to form a "double Intertropical convergence zone" and an equatorial cold tongue that extends too far westward. Such biases limit our confidence in projections of the future climate change for this region. In this study, we use a downscaling strategy based on a regional atmospheric general circulation model that accurately captures the SPCZ present-day climatology and interannual variability. More specifically, we investigate the sensitivity of the projected rainfall response to either just correcting present-day CMIP5 Sea Surface Temperature (SST) biases or correcting projected SST changes using an emergent constraint approach. While the equatorial western Pacific projected rainfall increase is robust in our experiments and CMIP5, correcting the projected CMIP5 SST changes yields a considerably larger reduction (~ 25%) than in CMIP5 simulations (~ + 3%) in the southwestern Pacific. Indeed, correcting the projected CMIP5 warming pattern yields stronger projected SST gradients, and more humidity convergence reduction under the SPCZ. Finally, our bias-corrected set of experiments yields an increase in equatorial rainfall and SPCZ variability in the future, but does not support the future increase in the frequency of zonal SPCZ events simulated by CMIP5 models. This study hence suggests that atmospheric downscaling studies should not only correct CMIP5 present-day SST biases but also projected SST changes to improve the reliability of their projections. Additional simulations with different physical parameterizations yield robust results.